Date of this version: April 1, 2004
A: (For rigorous definitions and theory, which are beyond the scope of this document, the interested reader is referred to the many LP textbooks in print, a few of which are listed in the references section.)
A Linear Program (LP) is a problem that can be expressed as follows (the so-called Standard Form):
minimize cx subject to Ax = b x >= 0where x is the vector of variables to be solved for, A is a matrix of known coefficients, and c and b are vectors of known coefficients. The expression "cx" is called the objective function, and the equations "Ax=b" are called the constraints. All these entities must have consistent dimensions, of course, and you can add "transpose" symbols to taste. The matrix A is generally not square, hence you don't solve an LP by just inverting A. Usually A has more columns than rows, and Ax=b is therefore quite likely to be under-determined, leaving great latitude in the choice of x with which to minimize cx.
The word "Programming" is used here in the sense of "planning"; the necessary relationship to computer programming was incidental to the choice of name. Hence the phrase "LP program" to refer to a piece of software is not a redundancy, although I tend to use the term "code" instead of "program" to avoid the possible ambiguity.
Although all linear programs can be put into the Standard Form, in practice it may not be necessary to do so. For example, although the Standard Form requires all variables to be non-negative, most good LP software allows general bounds l <= x <= u, where l and u are vectors of known lower and upper bounds. Individual elements of these bounds vectors can even be infinity and/or minus-infinity. This allows a variable to be without an explicit upper or lower bound, although of course the constraints in the A-matrix will need to put implied limits on the variable or else the problem may have no finite solution. Similarly, good software allows b1 <= Ax <= b2 for arbitrary b1, b2; the user need not hide inequality constraints by the inclusion of explicit "slack" variables, nor write Ax >= b1 and Ax <= b2 as two separate constraints. Also, LP software can handle maximization problems just as easily as minimization (in effect, the vector c is just multiplied by -1).
The importance of linear programming derives in part from its many applications (see further below) and in part from the existence of good general-purpose techniques for finding optimal solutions. These techniques take as input only an LP in the above Standard Form, and determine a solution without reference to any information concerning the LP's origins or special structure. They are fast and reliable over a substantial range of problem sizes and applications.
Two families of solution techniques are in wide use today. Both visit a progressively improving series of trial solutions, until a solution is reached that satisfies the conditions for an optimum. Simplex methods, introduced by Dantzig about 50 years ago, visit "basic" solutions computed by fixing enough of the variables at their bounds to reduce the constraints Ax = b to a square system, which can be solved for unique values of the remaining variables. Basic solutions represent extreme boundary points of the feasible region defined by Ax = b, x >= 0, and the simplex method can be viewed as moving from one such point to another along the edges of the boundary. Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region. These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in 1984.
The related problem of integer programming (or integer linear programming, strictly speaking) requires some or all of the variables to take integer (whole number) values. Integer programs (IPs) often have the advantage of being more realistic than LPs, but the disadvantage of being much harder to solve. The most widely used general-purpose techniques for solving IPs use the solutions to a series of LPs to manage the search for integer solutions and to prove optimality. Thus most IP software is built upon LP software, and this FAQ applies to problems of both kinds.
Linear and integer programming have proved valuable for modeling many and diverse types of problems in planning, routing, scheduling, assignment, and design. Industries that make use of LP and its extensions include transportation, energy, telecommunications, and manufacturing of many kinds. A sampling of applications can be found in many LP textbooks, in books on LP modeling systems, and among the application cases in the journal Interfaces.
A: Thanks to the advances in computing of the past decade, linear programs in a few thousand variables and constraints are nowadays viewed as "small". Problems having tens or hundreds of thousands of continuous variables are regularly solved; tractable integer programs are necessarily smaller, but are still commonly in the hundreds or thousands of variables and constraints. The computers of choice for linear and integer programming applications are Pentium-based PCs and the several varieties of Unix workstations.
There is more to linear programming than optimal solutions and number-crunching, however. This can be appreciated by observing that modern LP software comes in two related but very different kinds of packages:
Most modeling systems support a variety of algorithmic codes, while the more popular codes can be used with many different modeling systems. Because packages of the two kinds are often bundled for convenience of marketing or operation, the distinction between them is sometimes obscured, but it is important to keep in mind when attempting to sort through the many alternatives available.
Large-scale LP algorithmic codes rely on general-structure sparse matrix techniques and numerous other refinements developed through years of experience. The fastest and most reliable codes thus represent considerable development effort, and tend to be expensive except in very limited demonstration or "student" versions. Those codes that are free -- to all, or at least for research and teaching -- tend to be somewhat less robust, though they are still useful for many problems. The ability of a code to solve any particular class of problems cannot easily be predicted from problem size alone; some experimentation is usually necessary to establish difficulty.
Large-scale LP modeling systems are commercial products virtually without exception, and tend to be as expensive as the commercial algorithmic codes (again with the exception of small demo versions). They vary so greatly in design and capability that a description in words is adequate only to make a preliminary decision among them; your ultimate choice is best guided by using each candidate to formulate a model of interest.
Listed below are summary descriptions of available free codes, and a tabulation of many commercial codes and modeling systems for linear (and integer) programming. A list of free demos of commercial software appears at the end of this section.
Another useful source of information is the Optimization Software Guide by Jorge More' and Stephen Wright. It contains references to about 75 available software packages (not all of them just LP), and goes into more detail than is possible in this FAQ; see in particular the sections on "linear programming" and on "modeling languages and optimization systems." An updated Web version of this book is available on the NEOS Guide. Another good soruce of feature summaries and contact information is the Linear Programming Software Survey compiled by OR/MS Today (which also has the largest selection of advertisements for optimization software). Much information can also be obtained through the websites of optimization software developers, many of which are identified in the writeup and tables below.
To provide some idea of the relative performance of LP codes, a collection of benchmark results for optimization software is being compiled by Hans Mittelmann of Arizona State University. It includes tests of numerous simplex and interior-point codes for linear programming as well as branch-and-bound codes for linear integer programming; both commercial packages and downloadable free implementations are included. (Many nonlinear programming benchmarks are also available at this site.)
When evaluating any performance comparison, whether performed by a customer, vendor, or disinterested third party, keep in mind that all high-quality codes provide options that offer superior performance on certain difficult kinds of LP or IP problems. Benchmark studies of the "default settings" of codes will fail to reflect the power of any optional settings that are applicable.
These codes are not as fast or robust on average as the commercial products, but they're a a reasonable first try if you're not sure what level of power you need.
Based on the simplex method:
lp_solve is a downloadable code for linear and mixed-integer programming, currently maintained by Peter Notebaert (email@example.com). Version 4.0 is now available, under the Lesser GNU Public License, along with converters between MPS format and lp_solve's own input format.
GLPK (GNU Linear Programming Kit) is a C package that includes simplex (and also primal-dual interior point) methods for linear programming, a branch-and-bound implementation for integer programming, and a translator for the GNU MathProg language (a subset of AMPL). It is made available as open source under the GNU General Program License; Andrew Makhorin (firstname.lastname@example.org) is the developer and maintainter.
LP-Optimizer is a simplex-based code for linear and integer programs, written by Markus Weidenauer (email@example.com). Free Borland Pascal 7.0 and Borland Delphi 4 source is available for downloading, as are executables for DOS and for Windows (95 or later).
SoPlex is an object-oriented implementation of the primal and dual simplex algorithms, developed by Roland Wunderling. Source code is available free for research uses at noncommercial and academic institutions.
Among the SLATEC library routines is a Fortran sparse implementation of the simplex method, SPLP. Its documentation states that it can solve LP models of "at most a few thousand constraints and variables".
Based on interior-point methods:
The Optimization Technology Center at Argonne National Laboratory and Northwestern University has developed PCx, an interior-point code that is freely available for downloading. PCx is available in Fortran or C source or a variety of Windows and Unix executables, with an optional Java-based GUI interface. Input can be specified in MPS form or by use of the AMPL modeling language.
Csaba Meszaros (firstname.lastname@example.org) has written BPMPD, an interior-point code for linear and convex quadratic programs. A demonstration version, which solves problems of any size but does not report optimal values of the variables for problems larger than about 500 x 500, is available as a Windows95/NT executable or DLL. Separately, a large variety of Unix binaries for Linux and four workstation platforms are available for downloading.
Jacek Gondzio (email@example.com) has developed the interior-point LP (and convex QP) solver HOPDM. Several papers (also available at the HOPDM website) detail the features of this solver, which include automatic selection of multiple centrality correctors and factorization method, and a "warm start" feature for taking advantage of known solutions. A public-domain Fortran version (2.13, LP only) can be downloaded, and a newer C version (2.30) is available on request to the developer.
If you want to solve an LP without downloading a code to your own machine, you can execute many of these interior-point codes (as well as varied commercial LP codes) through the NEOS Server.
LPL is a mathematical modeling language for formulating and maintaining linear and mixed-integer programs. It is particularly notable for its ability to also handle a variety of logical constraints, which are translated symbolically into algebraic constraints using 0-1 variables. You can download the software and documentation free of charge.
The open source "scenario scripting language" IPAT-S includes support for linear programming problems, using the lp_solve library.
Other software of interest:
COIN, a Common Optimization INterface for Operations Research, is an "open source" repository established with support from IBM Corporation. Source code initially available includes a parallel branch-cut-price framework, a cut generation library, an implementation of the Volume Algorithm for fast approximate solutions to combinatorial problems, and an open solver interface layer.
ABACUS is a C++ class library that "provides a framework for the implementation of branch-and-bound algorithms using linear programming relaxations that can be complemented with the dynamic generation of cutting planes or columns" (branch-and-cut and/or branch-and-price). It relies on CPLEX, SoPlex, or Xpress-MP to solve linear programs. Further information is available from Stefan Thienel, firstname.lastname@example.org.
Various small-scale implementations are mainly intended for instructional purposes.
The Operations Research Laboratory at Seoul National University, Korea offers C source for large-scale Linear Programming software (both Simplex and Barrier) and for numerous more specialized optimization problems.
Will Naylor has a collection of software he calls WNLIB. Routines of interest include a dense-matrix simplex method for linear programming (with anti-cycling and numerical stability "hacks") and a sparse-matrix transportation/assignment problem routine. (WNLIB also contains routines pertaining to nonlinear optimization.)
A code known as lp is Mike Hohmeyer's C implementation of Raimund Seidel's O(d! n) time linear programming algorithm. It's reputed to be extremely fast in low dimensions (say, d < 10), so that it's appropriate for a variety of geometric problems, especially with very large numbers of constraints.
The next several suggestions are for codes that are severely limited by the dense vector and matrix algebra employed in their implementations; they may be OK for models with (on the order of) 100 variables and constraints, but it's unlikely they will be satisfactory for larger models. In the words of Matt Saltzman (email@example.com):
For Macintosh users there is relatively little available, but here are a few possibilities:
Stephen F. Gale (firstname.lastname@example.org) writes:
The following suggestions may represent low-cost ways of solving LPs if you already have certain software available to you.
If your models prove to be too difficult for free or add-on software to handle, then you may have to consider acquiring a commercial LP code. Dozens of such codes are on the market. There are many considerations in selecting an LP code. Speed is important, but LP is complex enough that different codes go faster on different models; you won't find a "Consumer Reports" article to say with certainty which code is THE fastest. I usually suggest getting benchmark results for your particular type of model if speed is paramount to you. Benchmarking can also help determine whether a given code has sufficient numerical stability for your kind of models.
Other questions you should answer: Can you use a stand-alone code, or do you need a code that can be used as a callable library, or do you require source code? Do you want the flexibility of a code that runs on many platforms and/or operating systems, or do you want code that's tuned to your particular hardware architecture (in which case your hardware vendor may have suggestions)? Is the choice of algorithm (Simplex, Interior-Point) important to you? Do you need an interface to a spreadsheet code? Is the purchase price an overriding concern? If you are at a university, is the software offered at an academic discount? How much hotline support do you think you'll need? There is usually a large difference in LP codes, in performance (speed, numerical stability, adaptability to computer architectures) and in features, as you climb the price scale.
Information on commercial systems is provided in two tables below. The first lists packages that are primarily algorithmic codes, and the second lists modeling systems. For a more extensive summary, take a look at the Linear Programming Software Survey in the August 1999 issue of OR/MS Today.
In the tables below, product names are linked to product or developer websites where known. Under "Platform" is an indication of common environments in which the code runs, with the choices being Microsoft Windows (PC), Macintosh OS (M), and Unix/Linux (U). The codes under "Features" are as follows:
S=Simplex Q=Quadratic B=Barrier G=General Nonlinear N=Network I=Integer/CombinatorialAll product information is subject to change, and some delay may occur before changes are reflected in this table. Consult the products' developers or vendors for definitive information.
|Features||Platform||Phone (+1)||E-mail address|
|CPLEX||SBINQ||PC U||800-367-4564 |
|FortMP||SBIQ||PC U||+44 email@example.com|
|HI-PLEX||S||PC U||+44 firstname.lastname@example.org|
|ILOG Opt Suite||SBINQ||PC U||800-367-4564 |
|LAMPS||SI||PC U||+44 email@example.com|
|LINDO API||SBI||PC Ufirstname.lastname@example.org|
|MOSEK||SBQG||PC U||+45 email@example.com|
|MPSIII||SIN||PC U||703-412-3201 |
|SAS/OR||SINQG||PC M U||919-677-8000|
|SCICONIC||SI||PC U||+44 firstname.lastname@example.org|
|Solver DLL Platform||SIQG||PC||888-831-0333 |
|SOPT||SBIQG||PC U||732-264-4700 |
|XA||SI||PC M Uemail@example.com|
|Xpress-MP||SBIQ||PC U||201-567-9445 |
|Platform||Phone (+1)||E-mail address|
|AMPL||PC U||see firstname.lastname@example.org|
|LP-TOOLKIT||PC U||+33 email@example.com|
|OPL Studio||PC U||800-367-4564 |
|Premium Solver Platform||PC||888-831-0333 |
|TOMLAB||PC U M||+46 21-804760 |
|What's Best!||PC U Mfirstname.lastname@example.org|
|Xpress-Mosel||PC U||201-567-9445 |
In a number of cases, the developers have also written books about these modeling systems. Demo software with a corresponding book can provide a convenient and inexpensive package for use in teaching or self-study.
Many developers are also willing to arrange for you to "borrow" copies of their full-featured versions for purposes of evaluation. Details vary, however, so you'll have to check with each vendor whose product you're interested in.
A: Integer LP models are ones whose variables are constrained to take integer or whole number (as opposed to fractional) values. It may not be obvious that integer programming is a very much harder problem than ordinary linear programming, but that is nonetheless the case, in both theory and practice.
Integer models are known by a variety of names and abbreviations, according to the generality of the restrictions on their variables. Mixed integer (MILP or MIP) problems require only some of the variables to take integer values, whereas pure integer (ILP or IP) problems require all variables to be integer. Zero-one (or 0-1 or binary) MIPs or IPs restrict their integer variables to the values zero and one. (The latter are more common than you might expect, because many kinds of combinatorial and logical restrictions can be modeled through the use of zero-one variables.)
For the sake of generality, the following disucssion uses the term MIP to refer to any kind of integer LP problem; the other kinds can be viewed as special cases. MIP, in turn, is a particular member of the class of combinatorial or discrete optimization problems. In fact the problem of determining whether a MIP has an objective value less than a given target is a member of the class of "NP-complete" problems, all of which are very hard to solve (at least as far as anyone has been able to tell). Since any NP-complete problem is reducible to any other, virtually any combinatorial problem of interest can be attacked in principle by solving some equivalent MIP. This approach sometimes works well in practice, though it is by no means infallible.
People are sometimes surprised to learn that MIP problems are solved using floating point arithmetic. Most available general-purpose large-scale MIP codes use a procedure called "branch-and-bound" to search for an optimal integer solution by solving a sequence of related LP "relaxations" that allow some fractional values. Good codes for MIP distinguish themselves primarily by solving shorter sequences of LPs, and secondarily by solving the individual LPs faster. (The similarities between successive LPs in the "search tree" can be exploited to speed things up considerably.) Even more so than with regular LP, a costly commercial code may prove its value if your MIP model is difficult.
Another solution approach known generally as constraint logic programming or constraint programming (CP) has drawn increasing interest of late. Having their roots in studies of logical inference in artificial intelligence, CP codes typically do not proceed by solving any LPs. As a result, compared to branch-and-bound they search "harder" but faster through the tree of potential solutions. Their greatest advantage, however, lies in their ability to tailor the search to many constraint forms that can be converted only with difficulty to the form of an integer program; their greatest success tends to be with "highly combinatorial" optimization problems such as scheduling, sequencing, and assignment, where the construction of an equivalent IP would require the definition of large numbers of zero-one variables. Notable constraint programming codes include CHIP, ECLiPSe, GNU Prolog, IF/Prolog, ILOG Solver, Koalog Constraint Solver, MOzart, and SICStus Prolog. Much more information can be found in the Constraints Archive, which contains the the comp.constraints newsgroup FAQ, pages of constraint-related pointers, source code for various systems, benchmarks, a directory of people interested in constraints, constraint bibliographies, and a collection of on-line papers.
The IP and CP approaches are not so far apart as they may seem, particularly now that each is being adapted to incorporate some of the strengths of the other. Some fundamental connections are described in [Chandru and Hooker] and [Hooker].
Whatever your solution technique, you should be prepared to devote far more computer time and memory to solving a MIP problem than to solving the corresponding LP relaxation. (Or equivalently, you should be prepared to solve much smaller MIP problems than LP problems using a given amount of computer resources.) To further complicate matters, the difficulty of any particular MIP problem is hard to predict (in advance, at least!). Problems in no more than a hundred variables can be challenging, while others in tens of thousands of variables solve readily. The best explanations of why a particular MIP is difficult often rely on some insight into the system you are modeling, and even then tend to appear only after a lot of computational tests have been run. A related observation is that the way you formulate your model can be as important as the actual choice of solver.
Thus a MIP problem with hundreds of variables (or more) should be approached with a certain degree of caution and patience. A willingness to experiment with alternative formulations and with a MIP code's many search options often pays off in greatly improved performance. In the hardest cases, you may wish to abandon the goal of a provable optimum; by terminating a MIP code prematurely, you can often obtain a high-quality solution along with a provable upper bound on its distance from optimality. A solution whole objective value is within some fraction of 1% of optimal may be all that is required for your purposes. (Indeed, it may be an optimal solution. In contrast to methods for ordinary LP, procedures for MIP may not be able to prove a solution to be optimal until long after they have found it.)
Once one accepts that large MIP models are not typically solved to a proved optimal solution, that opens up a broad area of approximate methods, probabilistic methods and heuristics, as well as modifications to B&B. See [Balas] which contains a useful heuristic for 0-1 MIP models. See also the brief discussion of Genetic Algorithms and Simulated Annealing in the Nonlinear Programming FAQ.
A major exception to this somewhat gloomy outlook is that there are certain models whose LP solution always turns out to be integer, assuming the input data is integer to start with. In general these models have a "unimodular" constraint matrix of some sort, but by far the best-known and most widely used models of this kind are the so-called pure network flow models. It turns out that such problems are best solved by specialized routines, usually based on the simplex method, that are much faster than any general-purpose LP methods. See the section on Network models for further information.
Commercial MIP codes are listed with the commercial LP codes and modeling systems above. The following are notes on some publicly available codes for MIP problems.
A: If you want to try out your code on some real-world linear programs, there is a very nice collection of small-to-medium-size ones, with a few that are rather large, popularly known as the Netlib collection (although Netlib consists of much more than just LP). Also on netlib is a collection of infeasible LP models. See the readme file for a listing and further information. The test problem files (after you uncompress them) are in a format called MPS, which is described in another section of this document. Note that, when you receive a model, it may be compressed both with the Unix compression utility (use uncompress if the file name ends in .Z) and with an LP-specific program (grab either emps.f or emps.c at the same time you download the model, then compile/run the program to undo the compression).
There is a collection of mixed integer (linear) programming (or MIP) models, called MIPLIB, housed at Rice University.
TSPLIB is a library of traveling salesman and related problems, including vehicle routing problems.
For network flow problems, there are some generators and instances collected at DIMACS. The NETGEN and GNETGEN generator can be downloaded from netlib. Generators and instances for multicommodity network flow problems are maintained by the Operations Research group in the Department of Computer Science at the University of Pisa.
The commercial modeling language GAMS comes with about 160 test models, which you might be able to test your code with. There are also pages containing pointers to numerous examples in AMPL, MPL, and OPL.
There is a collection called MP-TESTDATA available at Konrad-Zuse-Zentrum fuer Informations-technik Berlin (ZIB). This directory contains various subdirectories, each of which has a file named "index" containing further information. Indexed at this writing are: assign, cluster, lp, ip, matching, maxflow, mincost, set-parti, steiner-tree, tsp, vehicle-rout, and generators.
John Beasley of the Imperial College Management School maintains the OR-Library, which lists linear programming and over 3 dozen other categories of optimization test problems.
Finally, Martin Chlond's pages on Integer Programming in Recreational Mathematics provide a variety of challenges for both modelers and software.
A: MPS format was named after an early IBM LP product and has emerged as a de facto standard ASCII medium among most of the commercial LP codes. Essentially all commercial LP codes accept this format, but if you are using public domain software and have MPS files, you may need to write your own reader routine for this. It's not too hard. See also the comment regarding the lp_solve code, in another section of this document, for the availability of an MPS reader.
The main things to know about MPS format are that it is column oriented (as opposed to entering the model as equations), and everything (variables, rows, etc.) gets a name. The MIPLIB site provides a concise summary of MPS format, and a more detailed description is given in [Murtagh].
MPS is a very old format, so it is set up as though you were using punch cards, and is not free format. Fields start in column 1, 5, 15, 25, 40 and 50. Sections of an MPS file are marked by so-called header cards, which are distinguished by their starting in column 1. Although it is typical to use upper-case throughout the file (like I said, MPS has long historical roots), many MPS-readers will accept mixed-case for anything except the header cards, and some allow mixed-case anywhere. The names that you choose for the individual entities (constraints or variables) are not important to the solver; you should pick names that are meaningful to you, or will be easy for a post-processing code to read.
Here is a little sample model written in MPS format (explained in more detail below):
NAME TESTPROB ROWS N COST L LIM1 G LIM2 E MYEQN COLUMNS XONE COST 1 LIM1 1 XONE LIM2 1 YTWO COST 4 LIM1 1 YTWO MYEQN -1 ZTHREE COST 9 LIM2 1 ZTHREE MYEQN 1 RHS RHS1 LIM1 5 LIM2 10 RHS1 MYEQN 7 BOUNDS UP BND1 XONE 4 LO BND1 YTWO -1 UP BND1 YTWO 1 ENDATA
For comparison, here is the same model written out in an equation-oriented format:
Optimize COST: XONE + 4 YTWO + 9 ZTHREE Subject To LIM1: XONE + YTWO < = 5 LIM2: XONE + ZTHREE > = 10 MYEQN: - YTWO + ZTHREE = 7 Bounds 0 < = XONE < = 4 -1 < = YTWO < = 1 End
Strangely, there is nothing in MPS format that specifies the direction of optimization. And there really is no standard "default" direction; some LP codes will maximize if you don't specify otherwise, others will minimize, and still others put safety first and have no default and require you to specify it somewhere in a control program or by a calling parameter. If you have a model formulated for minimization and the code you are using insists on maximization (or vice versa), it may be easy to convert: just multiply all the coefficients in your objective function by (-1). The optimal value of the objective function will then be the negative of the true value, but the values of the variables themselves will be correct.
The NAME card can have anything you want, starting in column 15. The ROWS section defines the names of all the constraints; entries in column 2 or 3 are E for equality rows, L for less-than ( <= ) rows, G for greater-than ( >= ) rows, and N for non-constraining rows (the first of which would be interpreted as the objective function). The order of the rows named in this section is unimportant.
The largest part of the file is in the COLUMNS section, which is the place where the entries of the A-matrix are put. All entries for a given column must be placed consecutively, although within a column the order of the entries (rows) is irrelevant. Rows not mentioned for a column are implied to have a coefficient of zero.
The RHS section allows one or more right-hand-side vectors to be defined; most people don't bother having more than one. In the above example, the name of the RHS vector is RHS1, and has non-zero values in all 3 of the constraint rows of the problem. Rows not mentioned in an RHS vector would be assumed to have a right-hand-side of zero.
The optional BOUNDS section lets you put lower and upper bounds on individual variables (no * wild cards, unfortunately), instead of having to define extra rows in the matrix. All the bounds that have a given name in column 5 are taken together as a set. Variables not mentioned in a given BOUNDS set are taken to be non-negative (lower bound zero, no upper bound). A bound of type UP means an upper bound is applied to the variable. A bound of type LO means a lower bound is applied. A bound type of FX ("fixed") means that the variable has upper and lower bounds equal to a single value. A bound type of FR ("free") means the variable has neither lower nor upper bounds.
There is another optional section called RANGES that I won't go into here. The final card must be ENDATA, and yes, it is spelled funny.
Most modeling systems support a variety of algorithmic codes, while the more popular codes can be used with many different modeling systems. Because packages of the two kinds are often bundled for convenience of marketing or operation, the distinction between them is sometimes obscured, but it is important to keep in mind when sorting through the many possibilities. See under Commercial Codes and Modeling Systems elsewhere in this FAQ for a list of modeling systems available. There are no free ones of note, but many do offer free demo versions.
Common alternatives to modeling languages and systems include spreadsheet front ends to optimization, and custom optimization applications written in general-purpose programming languages. You can find a discussion of the pros and cons of these approaches in What Modeling Tool Should I Use? on the Frontline Systems website.
The source of infeasibility is often difficult to track down. It may stem from an error in specifying some of the constraints in your model, or from some wrong numbers in your data. It can be the result of a combination of factors, such as the demands at some customers being too high relative to the supplies at some warehouses.
Upon detecting infeasibility, LP codes typically show you the most recent infeasible solution that they have encountered. Sometimes this solution provides a good clue as to the source of infeasibility. If it fails to satisfy certain capacity constraints, for example, then you would do well to check whether the capacity is sufficient to meet the demand; perhaps a demand number has been mistyped, or an incorrect expression for the capacity has been used in the capacity constraint, or or the model simply lacks any provision for coping with increasing demands. More often, unfortunately, LP codes respond to an infeasible problem by returning a meaninglessly infeasible solution, such as one that violates material balances.
A more useful approach is to forestall meaningless infeasibilities by explicitly modeling those sources of infeasibility that you view as realistic. As a simple example, you could add a new "slack" variable on each capacity constraint, having a very high penalty cost. Then infeasibilities in your capacities would be signalled by positive values for these slacks at the optimal solution, rather than by a mysterious lack of feasibility in the linear program as a whole. Many modelers recommend the use of "soft constraints" of this kind in all models, since in reality many so-called constraints can be violated for a sufficiently high price. Modeling approaches that use such constraints have a number of names, most notably "goal programming" and "elastic programming".
Several codes include methods for finding an "irreducible infeasible subset" (IIS) of constraints that has no feasible solution, but that becomes feasible if any one constraint is removed. John Chinneck has developed MINOS(IIS), an extended version of the MINOS package that finds an IIS when the constraints have no feasible solution; a demonstration copy is available for downloading. There are also IIS finders in CPLEX, LINDO, OSL, and Xpress-MP, as well as Premium Solver Platform for Excel.
Methods also exist for finding an "IIS cover" that has at least one constraint in every IIS. A minimal IIS cover is the smallest subset of constraints whose removal makes the linear program feasible. Further details and references for a variety of IIS topics are available in papers by John Chinneck.
The software system ANALYZE carries out various other analyses to detect structures typically associated with infeasibility. (A bibliography on optimization modeling systems collected by Harvey Greenberg of the University of Colorado at Denver contains cross-references to over 100 papers on the subject of model analysis.)
There are a few free software packages specifically for multiple objective linear programming, including:
The folklore is that generally decomposition schemes take a long time to converge, so that they're slower than just solving the model as a whole -- although research continues. For now my advice, unless you are using OSL or your model is so huge that you can't buy enough memory to hold it, is to not bother decomposing it. It's probably more cost effective to upgrade your solver than to invest more time in programming (a good piece of advice in many situations).
Ken Clarkson has written Hull, an ANSI C program that computes the convex hull of a point set in general dimension. The input is a list of points, and the output is a list of facets of the convex hull of the points, each facet presented as a list of its vertices.
Qhull computes convex hulls as well as Delaunay triangulations, halfspace intersections about a point, Voronoi diagrams, and related objects. It uses the "Beneath Beyond" method, described in [Edelsbrunner].
Komei Fukuda's cdd solves both the convex hull and vertex enumeration problems, using the Double Description Method of Motzkin et al. There are also a C++ version (cdd+) and a C-library version (cddlib) that can be compiled to run with both floating-point and GMP exact rational arithmetics.
David Avis's lrslib is a self-contained ANSI C implementation of the reverse search algorithm for vertex enumeration/convex hull problems, implemented as a callable library with a choice of three arithmetic packages. VE041, another implementation of this approach by Fukuda and Mizukoshi, is available in a Mathematica implementation.
See also the directory of computational geometry software compiled by Nina Amenta; and the archive of the former University of Minnesota Geometry Center website, which includes links to downloadable software.
Other algorithms for such problems are described in [Swart], [Seidel], and [Avis]. Such topics are said to be discussed in [Schrijver] (page 224), [Chvatal] (chapter 18), [Balinski], and [Mattheis] as well.
The network linear programming problem is to minimize the (linear) total cost of flows along all arcs of a network, subject to conservation of flow at each node, and upper and/or lower bounds on the flow along each arc. This is a special case of the general linear programming problem. The transportation problem is an even more special case in which the network is bipartite: all arcs run from nodes in one subset to the nodes in a disjoint subset. A variety of other well-known network problems, including shortest path problems, maximum flow problems, and certain assignment problems, can also be modeled and solved as network linear programs. Details are presented in many books on linear programming and operations research.
Network linear programs can be solved 10 to 100 times faster than general linear programs of the same size, by use of specialized optimization algorithms. Some commercial LP solvers include a version of the network simplex method for this purpose. That method has the nice property that, if it is given integer flow data, it will return optimal flows that are integral. Integer network LPs can thus be solved efficiently without resort to complex integer programming software.
Unfortunately, many different network problems of practical interest do not have a formulation as a network LP. These include network LPs with additional linear "side constraints" (such as multicommodity flow problems) as well as problems of network routing and design that have completely different kinds of constraints. In principle, nearly all of these network problems can be modeled as integer programs. Some "easy" cases can be solved much more efficiently by specialized network algorithms, however, while other "hard" ones are so difficult that they require specialized methods that may or may not involve some integer programming. Contrary to many people's intuition, the statement of a hard problem may be only marginally more complicated than the statement of some easy problem.
A canonical example of a hard network problem is the "traveling salesman" problem of finding a shortest tour through a network that visits each node once. A canonical easy problem not obviously equivalent to a linear program is the "minimum spanning tree" problem to find a least-cost collection of arcs that connect all the nodes. But if instead you want to connect only some given subset of nodes (the "Steiner tree" problem) then you are faced with a hard problem. These and many other network problems are described in some of the references below.
Software for network optimization is thus in a much more fragmented state than is general-purpose software for linear programming. The following are some of the implementations that are available for downloading. Most are freely available for many purposes, but check their web pages or "readme" files for details.
The TSP has attracted many of the best minds in the optimization field, and serves as a kind of test-bed for methods subsequently applied to more complex and practical problems. Methods have been explored both to give proved optimal solutions, and to give approximate but "good" solutions, with a number of codes being developed as a result:
For practical purposes, the traveling salesman problem is only the simplest case of what are generally known as vehicle-routing problems. Thus commercial software packages for vehicle routing -- or more generally for "supply chain management" or "logistics" -- may have TSP routines buried somewhere within them. OR/MS Today published a detailed vehicle routing software survey in their August 2000 issue.
The packing problem can be regarded as a kind of cutting in reverse, where the goal is to fill large spaces with specified smaller pieces in the most economical (or profitable) way. As with cutting, there are one-dimensional problems (also called knapsack problems) and two-dimensional problems, but there are also many three-dimensional cases such as arise in filling trucks or shipping containers. The size measure is not always length or width; it may be weight, for example.
Except for some very special cases, cutting and packing problems are hard (NP-complete) like integer programming or the TSP. The simpler one-dimensional instances are often not hard to solve in practice, however:
There has been a great deal written on cutting and packing topics, but it tends to be scattered. You might want to start by looking at the web page of the Special Interest Group on Cutting and Packing and the "application-oriented research bibliography" in [Sweeney]. A search through some of the INFORMS databases will also turn up a lot of references. In fact, even an ordinary web search engine can find you a lot on this topic; try searching on "cutting stock".
Among the commercial systems are CutRite, act/cut and act/square of Alma, Cutting Optimizer of Ardis, Cutter of Escape, Cube-IQ and PlanIQ of MagicLogic, PLUS1D and PLUS2D of Nirvana Technologies, Cut Planner of Pattern Systems, and SmartTRIM of Strategic Systems. Particular application areas, from paper to carpeting, have also given rise to their own specialized cutting-stock tools, which can often be found by a web search on the area of interest.
[Thanks to Derek Holmes for the following text.] Your success solving a stochastic program depends greatly on the characteristics of your problem. The two broad classes of stochastic programming problems are recourse problems and chance-constrained (or probabilistically constrained) problems.
Recourse Problems are staged problems wherein one alteranates decisions with realizations of stochastic data. The objective is to minimize total expected costs of all decisions. The main sources of code (not necessarily public domain) depend on how the data is distributed and how many stages (decision points) are in the problem.
Chance-Constrained Programming (CCP) problems are not usually staged, and have a constraint of the form Pr( Ax <= b ) >= alpha. The solvability of CCP problems depends on the distribution of the data (A &/v b). I don't know of any public domain codes for CCP probs., but you can get an idea of how to approach the problem by reading Chapter 5 by Prof. A. Prekopa (email@example.com) Y. Ermoliev, and R. J-B. Wets, eds., Numerical Techniques for Stochastic Optimization (Series in Comp. Math. 10, Springer-Verlag, 1988).
Both Springer Verlag texts mentioned above are good introductory references to Stochastic Programming. The Stochastic Programming E-Print Series collects many recent papers in the field.
For a MIP model with both integer and continuous variables, you could get a limited amount of information by fixing the integer variables at their optimal values, re-solving the model as an LP, and doing standard post-optimal analyses on the remaining continuous variables; but this tells you nothing about the integer variables, which presumably are the ones of interest. Another MIP approach would be to choose the coefficients of your model that are of the most interest, and generate "scenarios" using values within a stated range created by a random number generator. Perhaps five or ten scenarios would be sufficient; you would solve each of them, and by some means compare, contrast, or average the answers that are obtained. Noting patterns in the solutions, for instance, may give you an idea of what solutions might be most stable. A third approach would be to consider a goal-programming formulation; perhaps your desire to see post-optimal analysis is an indication that some important aspect is missing from your model.
Interior-point methods for LP have entirely different requirements for a good starting point. Any reasonable interior-point-based LP code has its own routines for picking a starting point that is "well-centered" away from the constraints, in an appropriate sense. There is not much advantage to supplying your own starting point of any kind -- at least, not at the current state of the art -- and some codes do not even provide an option for giving a starting point.
The simplest answer to the problem of degeneracy/cycling is often to "get a better optimizer", i.e. one with stronger pricing algorithms, and a better selection of features. However, obviously that is not always an option (money!), and even the best LP codes can run into degeneracy on certain models. Besides, they say it's a poor workman who blames his tools.
So, when one cannot change the optimizer, it's expedient to change the model. Not drastically, of course, but a little "noise" can usually help to break the ties that occur during the Simplex method. A procedure that can work nicely is to add, to the values in the RHS, random values roughly six orders of magnitude smaller. Depending on your model's formulation, such a perturbation may not even seriously affect the quality of the solution values. However, if you want to switch back to the original formulation, the final solution basis for the perturbed model should be a useful starting point for a "cleanup" optimization phase. (Depending on the code you are using, this may take some ingenuity to do, however.)
Another helpful tactic: if your optimization code has more than one solution algorithm, you can alternate among them. When one algorithm gets stuck, begin again with another algorithm, using the most recent basis as a starting point. For instance, alternating between a primal and a dual method can move the solution away from a nasty point of degeneracy. Using partial pricing can be a useful tactic against true cycling, as it tends to reorder the columns. And of course Interior Point algorithms are much less affected by (though not totally immune to) degeneracy. Unfortunately, the optimizers richest in alternate algorithms and features also tend to be least prone to problems with degeneracy in the first place.
A: What follows is an idiosyncratic list, based on my own preferences, various people's recommendations on the net, and recent announcements of new publications. It is divided into the following categories:
Regarding the common question of the choice of textbook for a college LP course, it's difficult to give a blanket answer because of the variety of topics that can be emphasized: brief overview of algorithms, deeper study of algorithms, theorems and proofs, complexity theory, efficient linear algebra, modeling techniques, solution analysis, and so on. A small and unscientific poll of ORCS-L mailing list readers in 1993 uncovered a consensus that [Chvatal] was in most ways pretty good, at least for an algorithmically oriented class; of course, some new candidate texts have been published in the meantime. For a class in modeling, a book about a commercial code would be useful (LINDO, AMPL, GAMS were suggested), especially if the students are going to use such a code; and many are fond of [Williams], which presents a considerable variety of modeling examples.
Many optimization packages are distributed from their own Web sites. Numerous links to these sites are provided elsewhere in this FAQ, especially under the Where is there good software? question.
This article is Copyright © 2000 by Robert Fourer. It may be freely redistributed in its entirety provided that this copyright notice is not removed. It may not be sold for profit or incorporated in commercial documents without the written permission of the copyright holder. Permission is expressly granted for this document to be made available for file transfer from installations offering unrestricted anonymous file transfer on the Internet.
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If you wish to cite this FAQ formally -- this may seem strange, but it does come up -- you may use:
Robert Fourer (firstname.lastname@example.org), "Linear Programming Frequently Asked Questions," Optimization Technology Center of Northwestern University and Argonne National Laboratory, http://www-unix.mcs.anl.gov/otc/Guide/faq/ LinearProgrammingFAQ.htm (2000).Suggestions, corrections, topics you'd like to see covered, and additional material are all solicited. Send them to email@example.com.